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Tessellation

What are Tessellations and how are they used in Architecture?

Tessellations consist of a flat surface where one or more geometric shapes are utilized to form a pattern with no overlaps and gaps. Therefore, a tessellation is any kind of replicating form of

symmetrical and interlocking profiles. Tessellations are sometimes referred to as 'tilings'. According to Khaira, J., (2009.), the word tiling refers to a pattern of polygons (shapes with straight sides)

only. However, this is not the case as the use of both regular and irregular polygons can be adopted, which makes the resulting pattern more interesting to the human eye. The earliest and most

common tessellation found are triangles, squares and hexagons, as other irregular polygons are more difficult to assemble and to form. However, the more complex the plane is, the more

fascinating the surface becomes. This phenomena is an important area in mathematics, because, by mathematical computation it can be easily executed and manipulated to find use in art and

architecture. Architects and artists find this invention very intrigue to the human eye, therefore, this phenomena is extensively used in buildings walls as a main feature or for structural purpose. The

following images are examples of tessellation used in architecture as a feature wall or as the main structural component.

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Mathematics behind Tessellations : Which Shapes can Tessellate and Why can they do so?

As mentioned, the simplest form of tessellation is made up of regular polygons. Therefore, 'Regular Tessellations' are defined as being constructed

from only one kind of regular polygon. From and experiment conducted by Khaira, J., (2009.), it was concluded that, squares, equilateral triangles

and regular hexagons do form regular tessellations. However, even though pentagons and heptagons are considered as being regular polygons,

they cannot form a regular tessellation as they leave gaps or produce overlaps. This is where the mathematical part comes into play. In order for a

shape to tessellate, the interior angles must fill all of the space around a vertex, i.e., their interior angles must add up to 360 degrees (Khaira,

2009). Therefore, before starting to tessellate with any invented regular or irregular polygon, one must first calculate the interior angles. In order to

do this, one has to know that the exterior angles of both irregular and regular polygon adds up to 360 degrees. Once this is identified, one can than

deduce the interior angles. By summing them up one can then identify if they add up to 360 degrees. When the addition results in the answer being

360 degrees than one knows for sure that the shape will completely fill the space around a point. This point is called a vertex. According to

mathematical scientists, a vertex is defined as the point at which two or more straight lines meet, i.e. forming a corner. Therefore, once the space is

consumed, the shape will tessellate.

The above words can be summarized by the following equation to work out the angle of regular polygon: where ‘a’ is the interior angle and ‘n’ is the number of sides the polygon has. This is because

360 divided by the number of sides of the polygon gives the exterior angle, and when the exterior angle is subtracted from 180, we get the interior angle of the polygon.

In order to determine how many polygons are needed to fill the space around a vertex and allow the polygon to tessellate, another formula is used: where ‘k(n)’ is the number of polygons needed, ‘n’

is the number of sides and ‘a’ is the interior angle.

Therefore, it can be deduced that regular polygons that can fill the space around a vertex can tessellate. In more mathematical terms, regular polygons with interior angles that are a factor of 360,

can tessellate. Because of this, only regular polygons with 3, 4 or 6 sides - equilateral triangles, squares and regular hexagons - can perfectly fill 360° and tessellate by themselves.

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Symmetry and Transformations of Tilings

Symmetry is the process of taking a shape and through certain movements, matching

it exactly to another shape. This is the process how tessellation is created, by creating

the same motion a number of times. The technique of forming symmetry is

Transformational symmetry. Such transformations are , translations, rotations,

reflections and glide reflections. Such examples of these transformations are show in

the diagrams shown on the left hand side and bottom of this page.

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Simons Center for Geometry and Physics - State University of New York at Stony Brook

After having defined the meaning of Tessellation and the Mathematics behind such phenomena, I shall discuss the

application to its use in architecture regarding the Simons Centre for Geometry and Physics at the state University of New

York. The Simons Center is a facility for theoretical physics and mathematics. The focus of the center is mathematical

physics and the interface of geometry and physics. It was founded in 2007 by a gift from the James and Marilyn Simons

Foundation, but what interests me is not the buildings itself but the Tessellated facade that is found in this structure. This

facade is installed with an engineered kinetic tessellated surface known as TessellateTM. In fact Simons Center was the first

ever building to install the TessellateTM. The installation was completed on 2, November 2010, which serves as the building's

artistic centre piece and as a functional piece of shading integrated within its south facing glass facade. The kinetic façade

system can physically adapt to changes in daylight, solar gain, airflow and privacy by altering their configuration. The pattern

comprises a series of tessellated pattern such as hexagons, squares, triangles and circles. As these patterns align and

diverge, the visual effect is of sparse geometric patterns that blossom into an opaque mesh. The result is a kinetic surface

that spans 38 square meters and imbues the building with the functional capacity to dynamically change its opacity.

Tessellation TM

Tessellate™ is a self-contained, framed screen whose perforated pattern can continually shift and evolve to create a

dynamic architectural element that is able to regulate light and solar gain, ventilation and airflow, privacy, and views. It is

designed for walls, glass and glazing systems, and dividers as a versatile, multi -purpose façade. Tessellate™ consists of a

series of stacked panels that can be constructed of various metals or plastics. As these layers move and overlap, the result

is a kaleidoscopic visual display of patterns aligning and then diverging into a fine, light-diffusing mesh creating a living

environment. Tessellate is controlled using location-based sensory data to respond to light and weather conditions and

integrates into the building management system. For instance, when high levels of direct light are detected, the metal panels

diverge, and their patterns completely overlap, blocking the sun's rays. Tessellate™ is suitable for all building geometries.

Panel shape and size can be tailored to match the architect's building design and requirements. This product can be

constructed from non-rectangular profiles, to components comprising a three-dimensional surface. Since the problem of

organizing the space is achieved by mathematical equation, the engineers that produced Tessellate™ perforations, are

available in an endless range of patterns. Tessellate™ can be customized to fit any architectural vision. Mathematics has

made the options virtually endless.

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Perception of Space

The facility of this engineered facade is the shapes of the existing patterns as they are continuously shifting and overlapping each other, therefore the viewer's surrounding environment is

continuously changing. This has an effect of the space inside the building, i.e. there is a continuous change in play of light and shadows from this kinetic surface. As a result, one can have the

opportunity to perceive the same space differently due to the fact that light and shadows are regularly changing. Much like the effect of leaves that drops shadows onto the ground. This product can

at once create the same effect of speckled sunlight during the hot summer days while having the ability to allow direct sunlight in winter months. Solar shading as being the Tessellate's primary

function is an important consideration when designing an energy-efficient building envelope. It is a significant design challenge to reduce a building's energy consumption while maintaining daylight

levels that preserve views and promote a healthy internal environment. Tessellate's ability to shade adaptively addresses these concerns and improves a building's energy performance by

dynamically adjusting to environmental conditions and internal user preferences. According to Energy Environment Publishing, (2010.), by installing Tessellate™ to a south faced facade, the

consumption on the overall energy can be reduced by 6%. Therefore, fixed shading can reduce a building’s annual cooling load by 15%-20%. This creates a more natural and comfortable

environment to the visitor occupying the space within the structure. According to the same publish, "due to its automated, adaptive response to internal and external daylight levels, Tessellate™ can

provide 12-14% more hours per year when day-lighting systems can be used over artificial lights, as compared to fixed shading or traditional blinds." ( Energy Environment, 2010). When applied to a

glass façade Tessellate™ allows glass with a higher daylight transmission value and improved color rendering.

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3D Renders

Month - December

Time - 12:00

Since in winter the climate is not as warm as in summer days, the tessellated facade allows

more direct sunlight to enter the structure as the layers are overlapped onto each

other. This can be seen from the rendered image (left hand side) of the building's facade.

The shadows casted inside the building onto the floor are clear and the shapes are easily

defined. From this image one can notice the amount of light that this system allows to enters

without the need to use artificial lighting during the cold months.

Month - June

Time - 12:00

The Image on the right hand side is rendered to illustrate the difference between winter and

summer days. This shot is rendered in June as to show how the space changes when there is

direct sunlight hitting the south face of the building. The tessellated system, due to its sensors

installed, controls the movement of the shapes and by shifting them to one side, the amount of

sunrays entering the building is reduced. This helps to protect the visitors from excessive sunlight

while also allowing natural light to enter the space which helps the place to stay cooler. The

pattern that is produced on the floor is continuously changing with time as the patterns are

constantly shifting. This allows the users in the building to always experience a change in space,

making the environment more attractive to the user.

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3D Renders

Month - December and June

Time - 12:00

The following images shows the same aspect that the tessellated facade is

able to do both in winter (image on the left hand side) and in summer

(bottom image) from a different perspective. What is amazing about this

product is that from a mathematical equation, using mathematically proofed

shapes, a design can be invented to be used in architecture as an artistic

feature.

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AutoCad Drawings -

Shows how the pattern shifts when the TessellateTM senses a change in climatic temperature to control internal environment. One can notice how the

change in the pattern is perceived as being another design.

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Conclusion

In ancient and modern eras, the design of buildings has been influenced by mathematical ideas, such as the use of symmetry. Both historical and modern developments show that mathematics can

play an important role, ranging from appropriate descriptions of designs to guiding the designer's intuition. The use of geom etric shape to form a symmetric plane is a clear example of mathematical

relationships that is widely used in architecture. Over time, such phenomena have been defined as Tessellation. Tessellations are used extensively in architecture, both two-dimensional and three-

dimensional. Tessellations are easy to use in architecture, especially in two-dimensional, because even the simplest repeating pattern can look astonishing when it covers a large area. Modern

technology has improved over time and has moved into inventing and innovating a tessellated plan that can sense a change in climatic temperature, which has shifted to decrease the amount of

light entering the building.

TessellateTM - Simons Center for Geometry and Physics Installation

dimension and materials Adaptive Shading Coverage: 38 sq. meters

Material: Water jet-cut stainless steel, glass

Dimensions: 5.6m Wide x 6.7m Tall

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References:

1. Adaptive Building Initiative. (n.d.) . Simons Center for Geometry and Physics. Retrieved from: http://www.adaptivebuildings.com/simons-center.html

2. Aslaksen, H., (n.d.). Mathematics in Art and Architecture. Polygons and Tilings. Retrieved from: http://www.math.nus.edu.sg/aslaksen/teaching/maa/tilings.pdf

3. Energy Environment, (15 November, 2010.). Zahner and ABI Unveil Tessellate Adaptive Solar Shading Façade System. Retrieved from: http://www.china-aircon.com/detail-10009242/zahner-and-abi-unveil-tessellate-adaptive-solar-shading-fa-ade-system.html

4. Khaira J. ( 13 November, 2009.).What are Tilings and Tessellations and how are they used in Architecture?. Young Scientists J [serial online] 2009 [cited 2014 May 21];2:35-46. Retrieved

from: http://www.ysjournal.com/text.asp?2009/2/7/35/57766

5. TessellateTM Adaptive Facade System. Retrieved from: http://www.tessellatesurface.com/

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